GUNNAR NORDSTRÖM (1881-1923)

ON GRAVITATION AND RELATIVITY

Presented at the XVIIth International Congress of History of Science
University of California, Berkeley, July 31 - August 8, 1985.

Gunnar Nordström

Gunnar Nordström was certainly not the only physicist who made an attempt to formulate special
relativistic theory of gravitation. He is distinguished by having been the first, and also the most
successful at an attempt that was doomed to fail.

In 1916, when the only physical argument in favour of general relativity was the explanation
of the perihelion advance of the planet Mercury, Nordström gave up his own, then still current
theory in favour of Einstein's. His seemingly abrupt change of mind is a result of ten years of
thinking in terms of Minkowskiian electrodynamics, and getting no further with it.

Gunnar Nordström was a contemporary of Albert Einstein. Both men took a circuitous
path to university careers. During Einstein's early years at the Berne patent office, Nordström
became an engineer. Never a very practically oriented man, he turned into studying physical
chemistry. He arrived in Göttingen in April 1906 for about one year to study chemistry under
Walther Nernst. In Göttingen, the young Nordström became a wholehearted believer in relativity
in its Minkowskiian formulation. After having published only one paper in chemistry,
Nordström's whole remaining published work was focused almost exclusively in issues of
relativity, electrodynamics and gravitation.

Nordström's first major work was his dissertation on the energy equation for the
electromagnetic field of moving bodies. It was presented in 1908 at the University of Helsinki,
This work is an exposition of the Lorentz theory of electromagnetic phenomena as it was
formulated by Max Abraham in his Theory of Electricity. The Minkowskiian electrodynamics,
along with the theories of Maxwell and Hertz and the theory of Cohn were discussed at a separate
chapter, with no reference yet to Einstein.

Nordström's role in Helsinki was that of an introducer of novel physical ideas. He
established himself as a "Dozent" of theoretical physics at the university, at the same time teaching
elementary physics to gym students. Between years 1916 and 1918, Nordström worked in
Leiden, Holland. In 1918 he became a professor first of physics, then of mechanics at the Helsinki
University of Technology. During his whole career, his Scientific colleagues were Central
European, for no tradition of theoretical physics existed in Helsinki prior to Nordström. The
Finnish intellectual climate of the 1910s is reflected in one academicians retort to a request for
travel funds by Nordström: "One can study the fourth dimension at home, without any trips
abroad." Meanwhile, Nordström had already tried to introduce a fifth dimension to the fabric of
space-time in one of the very first attempts at a unified theory of electromagnetism and
gravitation.

In 1909 Nordström published a lengthy semi-popular article on relativity for the Finnish
Scientific audience: "Space and Time According to Einstein and Minkowski". There he at once
recognizes the basic meaning of Einstein's formulation of the principle of relativity. He also states
that Minkowski s "world postulate" is essentially identical with Einstein's principle of relativity,
although there exists a formal difference. By now, Nordström had become fascinated by the
Minkowskiian vision. He claimed that Einstein's version was designed for optical phenomena in
ether, while Minkowski's version was generally valid for mechanics as well as for electromagnetic
phenomena. He further commented that the Minkowskiian treatment for swiftly moving bodies
differed from that by Lorentz. Nordström's argument seems to have been based in considerations
of the principle of conservation of energy and basic concepts connected with it.

Having became an ardent supporter of Minkowskiian relativity, Nordström went on to
argue in an article published in Physikalische Zeitschrift in 1909 that the expression of electromagnetic ponderomotive force by Max Abraham was in contradiction with conservation of
energy. Abraham has given a finishing touch to the structure of classical electrodynamics. He
was also one of the very last eminent anti-relativists. Abraham was certainly a fair critic. He was
able to point out inconsistencies in relativists' work and to straighten them out while advocating
his own scepticismm. In his reply to Nordström, Abraham convinced his opponent by using
strictly relativistic arguments. The critical issue was the time dependence of mass in motion and
the consequent, interpretation of the energy equation from the point of view of relativistic
thermodynamics. Abraham's paper contains one of the first satisfactory formulations of the
energy-momentum tensor first conceptualized by Max Planck. Nordström closed the discussion
in 1911 by adopting Abraham's version of the ponderomotive force.

Abraham was soon to engage in another debate. In 1911, Einstein began to publish on
gravitation after a four-year-long silence. His early papers on scalar gravitation with a variable
velocity of light inspired Abraham both to attack relativity and to believe that even Einstein
himself had now abandoned his own basic principles. Abraham designed a scalar theory of
gravitation which wasneither relativistic nor consistent. Einstein abandoned scalar theories
altogether and began his long struggle with tensor theory.

Nordström's response to both was to suggest a scalar theory of gravitation in a rigid
relativistic sense of gravitation in a rigid relativistic sense. Nordström required the constancy of
the velocity of light in all frames of reference. Adopting Abraham's modification of the Poisson
equation, Nordström arrived at a potential dependence of gravitational mass. This first version of
the Nordström theory was sent to Physikalische Zeitschrift in October 1912. The theory was
notably imperfect in its definition of the source of the gravitational field, which had to be later
defined as the trace ofthe energy-momentum tensor as pointed out by Max von Laue in his
Relativitätsprinzip of 1911. Nordström made this important modification in his theory in summer
1913 while staying in Zurich in collaboration with Einstein. In September 1913 Einstein gave a
progress report on his work to the German Naturforscherversammlung, summarizing
Nordström's theory at a great length alongside with his own tensor theory.

The meaning of Nordström's scalar theory of gravitation is most clearly seen from its
covariant formulation which was made in 1914 by Einstein and A. D. Fokker. This formulation
was in effect a by-product of Einstein's own theory. It is in the footnote of this very paper where
Einstein raises his former objection to the existence of generally covariant field equations of
gravitation.

In terms of the formulation given by Einstein and Fokker, the Nordström theory can now
be expressed in terms of a conformally flat line element:

with

The equations of motion can be derived from this line element with a variational principle

The field equations are found by equating the curvature scalar R with the trace of the energy-momentum tensor, multiplied by a constant:

This was the first time when the complex tensor machinery gave Einstein generally
covariant equations. With this in mind he surely gained additional confidence for continuing his
own chosen path which led to the final formulation of the field equations of the general theory of
relativity in November 1915. The form of these equations is strikingly analogous with the scalar
field equation:

Nordström did not at once see the advantage of this approach. He proceeded by trying to
unify the known physical theories by treating the four-dimensional space-time as a projection of a
five-dimensional continuum. in practice, this attempt to rewrite the field equations of
electromagnetism and of gravitation in a mathematically analogous way by introducing a fifth
dimension remained only a mathematical trick. Nordström concluded that the predictions of this
new theory were either wrong or irrelevant. The theory is, however, interesting as an early scalar
version of the ten years later Kaluza-Klein theory.

In 1917 Max von Laue published a comprehensive exposition of the Nordström theory. As
late as over a year after the completion of general relativity, a scientist of Laue's standing could
still consider the Nordström theory as a serious rival to the Einstein theory. Of course, the
confirmation of the bending of light rays during the solar eclipse of 1919 changed all that.
Nordström's theory predicted no deflection at all, because in his theory electromagnetism is not
coupled to gravitational phenomena because the trace of the electromagnetic energy-momentum
tensor vanishes.

Nordström himself had already abandoned his own theory in 1916, despite the fact that the
question of bending of light still remained open, and despite the fact that Nordström had never
been very worried about the other serious flaw in the predictions of scalar gravitation, namely the
perihelion shift of the planet Mercury, which was of the wrong magnitude and even in the wrong
direction.

A major factor in Nordström's conversion to the general theory of relativity was his
growing consciousness of the importance of Hamilton's Principle for physics. After arriving in
Leiden in 1916, Nordström entered a first-rate community of theoretical physicists. He soon
began working with Einstein's new ideas on the Hamiltonian formulation of general relativity.
Nordström first considered the case of mechanics of continua, studied by Gustav Herglotz for
special relativistic cases, treating it first from the point of view of his own theory, then from that
of Einstein's This work led Nordström to study the general problem of a finite
matter source with an electric charge, which had already been considered by Hans Reissner and
by Hermann Weyl for a point source. This significant contribution by Nordström can be found in
modern textbooks of general relativity as the Reissner-Nordström metric for a nonrotating charge
distribution.

After 1918, Gunnar Nordström disappeared from the international scientific arena. Instead
of entering the competition for an assistant professorship in Berlin under Max Planck, he chose to
return to Helsinki with his new Dutch wife, Cornelia van Leeuwen, a physics student of Lorentz's.
Einstein and Max Born were helpful to the young couple when visas to Finland were urgently
needed. Another physicist who provided help to Nordström was Niels Bohr who forwarded his
mail during the war years by simply changing the envelopes.

Once in Helsinki, Nordström continued his work until his early death at an early age of 42 in
1923. Experimenting with radioactive substances had been one of his hobbies and very likely a
cause of his illness. As a professor at Helsinki University of Technology, Nordström twice
nominated Einstein for the Nobel prize in physics for his special and general theories of relativity.
He also taught a course of general relativity to Finnish students, which inspired later scientists to
continue the research begun by Nordström.

Nordström's scalar theory of gravitation was rediscovered in the 1960's when Robert Dicke,
A. L. Harvey and others used scalar models in discussing basic concepts connected with theories
of gravitation. Despite the fact that Nordström was wrong about gravitation, his theory still
serves as a stepping stone to understanding Einstein.